We define an algebraic group comprising symmetric chain complexes which
captures the first two stages of the Cochran-Orr-Teichner solvable filtration
of the knot concordance group in a single invariant. To achieve this we impose
additional structure on each chain complex which puts extra control on the
fundamental groups, and in particular on the way in which they can change in a
concordance.Comment: 51 pages, 2 figures. This a considerably shortened version of
  arXiv:1109.0761, to appear in Algebraic and Geometric Topolog

Blanchfield

Cappell

Casson

Kirby

Levine

Mark Powell

Ranicki

Stenström

English

arXiv

Crossref

Algebraic & Geometric Topology

A second order algebraic knot concordance group

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A Second Order Algebraic Knot Concordance Group

Let C be the topological knot concordance group of knots S 1 ⊂ S 3 under connected sum modulo slice knots. Cochran, Orr and Teichner defined a filtration: C ⊃ F (0) ⊃ F (0.5) ⊃ F (1) ⊃ F (1.5) ⊃ F (2) ⊃ ... The quotient C/F (0.5) is isomorphic to Levine's algebraic concordance group; F (0.5) is the algebraically slice knots. The quotient C/F (1.5) contains all metabelian concordance obstructions. Using chain complexes with a Poincaré duality structure, we define an abelian group AC 2, our second order algebraic knot concordance group. We define a group homomorphism C → AC 2 which factors through C/F (1.5), and we can extract the two stage Cochran-Orr-Teichner obstruction theory from our single stage obstruction group AC 2. Moreover there is a surjective homomorphism AC 2 → C/F (0.5), and we show that the kernel of this homomorphism is nontrivial

Powell, Mark

Enlighten

Let Knots be the abelian monoid of isotopy classes of knots S1 ⊂ S3 under connected sum, and let C be the topological knot concordance group of knots modulo slice knots. Cochran-Orr-Teichner [COT03] defined a filtration of C:
C ⊃ F(0) ⊃ F(0.5) ⊃ F(1) ⊃ F(1.5) ⊃ F(2) ⊃ . . .The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group AC1 [Lev69]; F(0.5)
is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance
obstructions. The Cochran-Orr-Teichner (1.5)-level two stage obstructions map the concordance
class of a knot to a pointed set (COT (C/1.5),U).
We define an abelian monoid of chain complexes P, with a monoid homomorphism Knots →
P. We then define an algebraic concordance equivalence relation on P and therefore a group
AC2 := P/ ~, our second order algebraic knot concordance group. The results of this thesis can
be summarised in the following diagram: . That is, we define a group homomorphism C → AC2 which factors through C/F(1.5). We can
extract the two stage Cochran-Orr-Teichner obstruction theory from AC2: the dotted arrows
are morphisms of pointed sets. Our second order algebraic knot concordance group AC2 is a
single stage obstruction group

Powell, Mark Andrew

Edinburgh Research Archive

Second order algebraic knot concordance group

First published in Algebraic & Geometric Topology in 12(2), published by Mathematical Sciences Publishers.Let C be the topological knot concordance group of knots $S^{1} \subset S^{3}$ under connected sum modulo slice knots. Cochran, Orr and Teichner defined a filtration: \[C \supset F_{(0)} \supset F_{(0.5)} \supset F_{(1)} \supset F_{(1.5)} \supset F_{(2)} \supset\cdots\]
The quotient $C/F_{(0.5)}$ is isomorphic to Levine’s algebraic concordance group; $F_{(0.5)}$ is the algebraically slice knots. The quotient $C/F_{(1.5)}$ contains all metabelian concordance obstructions.

Using chain complexes with a Poincaré duality structure, we define an abelian group  $AC_{2}$, our second order algebraic knot concordance group. We define a group homomorphism $C \rightarrow AC_{2}$ which factors through $C/F_{(1.5)}$, and we can extract the two stage Cochran–Orr–Teichner obstruction theory from our single stage obstruction group $AC_{2}$. Moreover there is a surjective homomorphism $AC_{2} \rightarrow C/F_{(0.5)}$, and we show that the kernel of this homomorphism is nontrivial

Powell, M.

IUScholarWorks (Indiana University)

A second order algebraic knot concordance group

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